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 A322038 Irregular triangle read by rows: for n >= 0, row n gives the coordination sequence for the tiling of a flat torus by a square grid with n points along each circuit. 4

%I

%S 1,1,1,2,1,1,4,4,1,4,6,4,1,1,4,8,8,4,1,4,8,10,8,4,1,1,4,8,12,12,8,4,1,

%T 4,8,12,14,12,8,4,1,1,4,8,12,16,16,12,8,4,1,4,8,12,16,18,16,12,8,4,1,

%U 1,4,8,12,16,20,20,16,12,8,4

%N Irregular triangle read by rows: for n >= 0, row n gives the coordination sequence for the tiling of a flat torus by a square grid with n points along each circuit.

%C More precisely, this is the coordination sequence for the quotient graph Z^2 / (nZ X nZ). The graph has n^2 vertices.

%C There are obvious generalizations: for example, Z^2 / (mZ X nZ) where m and n are not necessarily equal.

%H N. J. A. Sloane, <a href="/A322038/a322038_1.png">Illustration showing the tilings and coordination sequences for n = 4 and 5</a>

%F Since the underlying graphs are finite, the coordination sequences are polynomial P_n(x).

%F For n even, P_n(x) = (1+x)^2*(Sum_{i=0..(n-2)/2} x^i)^2;

%F for n odd, P_n(x) = (1 + 2*Sum_{i=0..(n-1)/2} x^i)^2.

%e The triangle begins:

%e 1,

%e 1,

%e 1, 2, 1,

%e 1, 4, 4,

%e 1, 4, 6, 4, 1,

%e 1, 4, 8, 8, 4,

%e 1, 4, 8, 10, 8, 4, 1,

%e 1, 4, 8, 12, 12, 8, 4,

%e 1, 4, 8, 12, 14, 12, 8, 4, 1,

%e 1, 4, 8, 12, 16, 16, 12, 8, 4,

%e 1, 4, 8, 12, 16, 18, 16, 12, 8, 4, 1,

%e ...

%Y The rows converge to A008574.

%K nonn,tabf

%O 0,4

%A _N. J. A. Sloane_, Dec 01 2018

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Last modified November 27 09:19 EST 2021. Contains 349365 sequences. (Running on oeis4.)